Generator

Bayes' theorem as two rectangles

A generator in the probability library, called 36 times across 6 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

bayes is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

Bayes' theorem as two rectangles. A unit square split by how common the condition is (1.0%) and then by how the test behaves. Of everyone who tests positive, the fraction who have it is 16.7%.

The same test, at every base rate

Evidence as steps on a scale of decibans

What a positive and a negative are each worth

Six orders of the same evidence

Two positive tests whose errors are shared

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

a halfer told it is the first day moves above one half ×1
a positive result is evidence for and a negative against ×1
a prior with thinner tails has a finite expected amount ×1
a thirder told it is the first day comes back to one half ×1
adding decibels is multiplying odds ×1
and at seven days it is 13/27 ×1
and at three doors it is two thirds ×1
and it is above a half for every pair ×1
and the boy–boy block holds 2d − 1 of them ×1
at every middle amount the other envelope is worth 5/4 as much on average ×1
at one awakening every account agrees on one half ×1
averaged over what might be seen, switching gains exactly nothing ×1
Broome's partial sums grow without bound ×1
each test alone keeps its false-positive rate ×1
every order of the evidence ends at the same odds ×1
everybody has it, so everybody positive has it ×1
nobody has it, so nobody who tests positive has it ×1
one of the two strategies wins, and only one ×1
one ticket a run is fair at one half ×1
one ticket an awakening is fair at 1/(wakes + 1) ×1
one to five positive likelihood ratios ×1
prior is a probability, between 0 and 1 ×1
seeing 1, the other is surely 2 ×1
seeing any larger amount, the other is double with probability 2/5 ×1
so its expected value is 1.1 times the amount seen ×1
specificity is a probability, between 0 and 1 ×1
staying wins as often as the first pick was right ×1
switching gets better with more doors ×1
switching wins two thirds against a host who knows, and half against one who does not ×1
tails brings between 1 and 8 awakenings ×1
the answer falls as the detail becomes commoner ×1
the answer rises with the base rate ×1
the asked families settle near 1/3 ×1
the awakenings settle near one share in wakes + 1 ×1
the chance of the larger is a half plus half the chance the threshold falls between the two ×1
the count is the formula (2 − p)/(4 − p) ×1
the curve agrees with the grid count ×1
the error rate is near Wald's bound 1/(1 + 10^(threshold/10)) ×1
the formula is the kept area ×1
the four cells fill the square ×1
the game needs at least three doors ×1
the halfer's square gives heads one half ×1
the ignorant host's story throws two of the six worlds away ×1
the ignorant-host variant is only drawn at three doors, where switching has one meaning ×1
the kept area gives the protocol's answer ×1
the largest pair is 2^N and 2^(N+1), N between 2 and 10 ×1
the marked row and column share one cell ×1
the met-a-child families settle near 1/2 ×1
the more the errors are shared, the less the second test adds ×1
the only news drawn is that it is the first day ×1
the paths differ on the way ×1
the question is any, older or met ×1
the runs settle near one half ×1
the shaded fraction is Bayes' theorem ×1
the thirder's square gives heads one share in wakes + 1 ×1
the trait takes between 1 and 12 values ×1
the two curves are mirror images about one half ×1
the two strategies exhaust the possibilities ×1
the view is one of sweep, monty, monty-scale, children, tuesday, trait, mention, children-sim, beauty, beauty-sim, beauty-bets, beauty-many, oddsladder, oddsorders, oddsdependent, oddssequential, oddsworth, envbounded, envbroome, envexpect, envsim, envthreshold ×1
the worlds fill the square ×1
with no shared cause the product is exact ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Probability

Bayes' theorem is a picture of a square

A test that is 99% accurate returns a positive result. The chance it is right can easily be under one in five, and the reason is visible the moment the population is drawn as a square rather than described as a formula.

Probability

Evidence measured in decibans

Write a probability as odds and take the logarithm, and every piece of evidence becomes a length. A positive result on a good test is thirteen decibans; a negative one is minus twenty. Lay the lengths end to end from the prior and the posterior is where they stop, in any order. The rule fails in exactly one way — when two pieces of evidence share a cause — and Turing built a code-breaking method on the arithmetic.

Probability

One coin, counted by runs and by wakings

Beauty is put to sleep and a fair coin is tossed. Heads, she is woken once; tails, twice, with the first waking erased from her memory. Each time she wakes she is asked how likely heads is. One half, say some; one third, say others; and unlike every earlier puzzle of this kind, stating the protocol exactly does not end the argument.

Probability

The door that was not opened

Three doors, one prize, a host who opens a losing door and offers a swap. Switching wins two times in three, and the reason is not about doors — it is about what the host was allowed to do.

Probability

The envelope that always looks better

Two envelopes, one holding twice as much as the other. Open one, see an amount, and reason that the other holds double or half with equal chance — so switching gains a quarter on average. By symmetry the same argument says switch back. The step that fails is not the arithmetic; it is the claim that double and half are equally likely whatever amount is seen, which no honest prior allows — and there is one prior under which the other envelope really does look better at every amount.

Probability

Two children and the sentence about one of them

A family has two children and at least one is a boy. The chance that both are boys is one in three — or one in two, or anything from one in three to certainty — and every one of those answers is right for some way the sentence could have come to be said. There is no host and no door, and the protocol is still the whole problem.

The whole library · What the figures prove